Scientific law

Continuity and transfer

Conservation laws can be expressed using the general continuity equation (for a conserved quantity) can be written in differential form as:

${\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {J} }$

where ρ is some quantity per unit volume, J is the flux of that quantity (change in quantity per unit time per unit area). Intuitively, the divergence (denoted ∇•) of a vector field is a measure of flux diverging radially outwards from a point, so the negative is the amount piling up at a point; hence the rate of change of density in a region of space must be the amount of flux leaving or collecting in some region (see the main article for details). In the table below, the fluxes flows for various physical quantities in transport, and their associated continuity equations, are collected for comparison.

Physics, conserved quantity	Conserved quantity q	Volume density ρ (of q)	Flux J (of q)	Equation
Hydrodynamics, fluids	m = mass (kg)	ρ = volume mass density (kg m−3)	ρ u, where u = velocity field of fluid (m s−1)	${\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot (\rho \mathbf {u} )}$
Electromagnetism, electric charge	q = electric charge (C)	ρ = volume electric charge density (C m−3)	J = electric current density (A m−2)	${\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {J} }$
Thermodynamics, energy	E = energy (J)	u = volume energy density (J m−3)	q = heat flux (W m−2)	${\displaystyle {\frac {\partial u}{\partial t}}=-\nabla \cdot \mathbf {q} }$
Quantum mechanics, probability	P = (r, t) = ∫|Ψ|2d3r = probability distribution	ρ = ρ(r, t) = |Ψ|2 = probability density function (m−3), Ψ = wavefunction of quantum system	j = probability current/flux	${\displaystyle {\frac {\partial |\Psi |^{2}}{\partial t}}=-\nabla \cdot \mathbf {j} }$

More general equations are the

Laws of classical mechanics

Principle of least action

Classical mechanics, including

where ${\displaystyle {\mathcal {S}}}$ is the action; the integral of the Lagrangian

${\displaystyle L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {\dot {q}} ,t)-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)}$

of the physical system between two times t₁ and t₂. The kinetic energy of the system is T (a function of the rate of change of the configuration of the system), and potential energy is V (a function of the configuration and its rate of change). The configuration of a system which has N degrees of freedom is defined by generalized coordinates q = (q₁, q₂, ... q_N).

There are generalized momenta conjugate to these coordinates, p = (p₁, p₂, ..., p_N), where:

${\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}}$

The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the generalized coordinates in the configuration space, i.e. the curve q(t), parameterized by time (see also parametric equation for this concept).

The action is a

Special relativity

The two postulates of special relativity are not "laws" in themselves, but assumptions of their nature in terms of relative motion.

They can be stated as "the laws of physics are the same in all

this replaces the Galilean transformation law from classical mechanics. The Lorentz transformations reduce to the Galilean transformations for low velocities much less than the speed of light c.

The magnitudes of 4-vectors are invariants - not "conserved", but the same for all inertial frames (i.e. every observer in an inertial frame will agree on the same value), in particular if A is the four-momentum, the magnitude can derive the famous invariant equation for mass–energy and momentum conservation (see invariant mass):

${\displaystyle E^{2}=(pc)^{2}+(mc^{2})^{2}}$

in which the (more famous) mass–energy equivalence E = mc² is a special case.

General relativity

General relativity is governed by the

Kepler's Laws, though originally discovered from planetary observations (also due to Tycho Brahe), are true for any central forces.^[16]

Newton's law of universal gravitation: For two point masses: ${\displaystyle \mathbf {F} ={\frac {Gm_{1}m_{2}}{\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} \,\!}$ For a non uniform mass distribution of local mass density ρ (r) of body of Volume V, this becomes: ${\displaystyle \mathbf {g} =G\int _{V}{\frac {\mathbf {r} \rho \,\mathrm {d} {V}}{\left|\mathbf {r} \right|^{3}}}\,\!}$	Gauss' law for gravity : An equivalent statement to Newton's law is: ${\displaystyle \nabla \cdot \mathbf {g} =4\pi G\rho \,\!}$
Kepler's 1st Law: Planets move in an ellipse, with the star at a focus ${\displaystyle r={\frac {\ell }{1+e\cos \theta }}\,\!}$ where ${\displaystyle e={\sqrt {1-(b/a)^{2}}}}$ is the eccentricity of the elliptic orbit, of semi-major axis a and semi-minor axis b, and ℓ is the semi-latus rectum. This equation in itself is nothing physically fundamental; simply the polar equation of an ellipse in which the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, where the orbited star is.
Kepler's 2nd Law: equal areas are swept out in equal times (area bounded by two radial distances and the orbital circumference): ${\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}={\frac {\left|\mathbf {L} \right|}{2m}}\,\!}$ where L is the orbital angular momentum of the particle (i.e. planet) of mass m about the focus of orbit,
Kepler's 3rd Law: The square of the orbital time period T is proportional to the cube of the semi-major axis a: ${\displaystyle T^{2}={\frac {4\pi ^{2}}{G\left(m+M\right)}}a^{3}\,\!}$ where M is the mass of the central body (i.e. star).

Thermodynamics

Laws of thermodynamics
First law of thermodynamics: The change in internal energy dU in a closed system is accounted for entirely by the heat δQ absorbed by the system and the work δW done by the system: ${\displaystyle \mathrm {d} U=\delta Q-\delta W\,}$ Second law of thermodynamics: There are many statements of this law, perhaps the simplest is "the entropy of isolated systems never decreases", ${\displaystyle \Delta S\geq 0}$ meaning reversible changes have zero entropy change, irreversible process are positive, and impossible process are negative.	Zeroth law of thermodynamics: If two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with one another. ${\displaystyle T_{A}=T_{B}\,,T_{B}=T_{C}\Rightarrow T_{A}=T_{C}\,\!}$ Third law of thermodynamics: As the temperature T of a system approaches absolute zero, the entropy S approaches a minimum value C: as T → 0, S → C.
For homogeneous systems the first and second law can be combined into the Fundamental thermodynamic relation: ${\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\,\!}$
Onsager reciprocal relations: sometimes called the Fourth Law of Thermodynamics ${\displaystyle \mathbf {J} _{u}=L_{uu}\,\nabla (1/T)-L_{ur}\,\nabla (m/T);\!}$ ${\displaystyle \mathbf {J} _{r}=L_{ru}\,\nabla (1/T)-L_{rr}\,\nabla (m/T).\!}$

• Newton's law of cooling
• Fourier's law
• Ideal gas law, combines a number of separately developed gas laws;
  • Boyle's law
  • Charles's law
  • Gay-Lussac's law
  • Avogadro's law, into one

now improved by other

equations of state

• Dalton's law (of partial pressures)
• Boltzmann equation
• Carnot's theorem
• Kopp's law

Electromagnetism

Maxwell's equations Gauss's law for electricity ${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}$ Gauss's law for magnetism ${\displaystyle \nabla \cdot \mathbf {B} =0}$ Faraday's law ${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$ Ampère's circuital law (with Maxwell's correction) ${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\ }$

Lorentz force law: ${\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}$

photons, see Maxwell's equations for details). They are modified in QED theory.

These equations can be modified to include magnetic monopoles, and are consistent with our observations of monopoles either existing or not existing; if they do not exist, the generalized equations reduce to the ones above, if they do, the equations become fully symmetric in electric and magnetic charges and currents. Indeed, there is a duality transformation where electric and magnetic charges can be "rotated into one another", and still satisfy Maxwell's equations.

Pre-Maxwell laws

These laws were found before the formulation of Maxwell's equations. They are not fundamental, since they can be derived from Maxwell's Equations. Coulomb's Law can be found from Gauss' Law (electrostatic form) and the Biot–Savart Law can be deduced from Ampere's Law (magnetostatic form). Lenz' Law and Faraday's Law can be incorporated into the Maxwell-Faraday equation. Nonetheless they are still very effective for simple calculations.

• Lenz's law
• Coulomb's law
• Biot–Savart law

Other laws

• Ohm's law
• Kirchhoff's laws
• Joule's law

Photonics

Classically, optics is based on a variational principle: light travels from one point in space to another in the shortest time.

• Fermat's principle

In

• Law of reflection
• Law of refraction, Snell's law

In physical optics, laws are based on physical properties of materials.

• Brewster's angle
• Malus's law
• Beer–Lambert law

In actuality, optical properties of matter are significantly more complex and require quantum mechanics.

Laws of quantum mechanics

Quantum mechanics has its roots in

• The state of a physical system, be it a particle or a system of many particles, is described by a
  wavefunction
  .

• Every physical quantity is described by an
  operator acting on the system; the measured quantity has a probabilistic nature
  .

• The
  wavefunction obeys the Schrödinger equation
  . Solving this wave equation predicts the time-evolution of the system's behavior, analogous to solving Newton's laws in classical mechanics.

These postulates in turn imply many other phenomena, e.g., uncertainty principles and the Pauli exclusion principle.

Quantum mechanics, Quantum field theory Schrödinger equation (general form): Describes the time dependence of a quantum mechanical system. ${\displaystyle i\hbar {\frac {d}{dt}}\left|\psi \right\rangle ={\hat {H}}\left|\psi \right\rangle }$ The Hamiltonian (in quantum mechanics) H is a self-adjoint operator acting on the state space, ${\displaystyle |\psi \rangle }$ (see Planck's constant .	Wave–particle duality Planck's constant , h). ${\displaystyle E=h\nu =\hbar \omega }$ De Broglie wavelength: this laid the foundations of wave–particle duality, and was the key concept in the Schrödinger equation, ${\displaystyle \mathbf {p} ={\frac {h}{\lambda }}\mathbf {\hat {k}} =\hbar \mathbf {k} }$ reduced Planck constant, similarly for time and energy ; ${\displaystyle \Delta x\,\Delta p\geq {\frac {\hbar }{2}},\,\Delta E\,\Delta t\geq {\frac {\hbar }{2}}}$ The uncertainty principle can be generalized to any pair of observables – see main article.
Wave mechanics Schrödinger equation (original form): ${\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi }$
Pauli exclusion principle: No two identical fermions can occupy the same quantum state (bosons can). Mathematically, if two particles are interchanged, fermionic wavefunctions are anti-symmetric, while bosonic wavefunctions are symmetric: ${\displaystyle \psi (\cdots \mathbf {r} _{i}\cdots \mathbf {r} _{j}\cdots )=(-1)^{2s}\psi (\cdots \mathbf {r} _{j}\cdots \mathbf {r} _{i}\cdots )}$ where ri is the position of particle i, and s is the spin of the particle. There is no way to keep track of particles physically, labels are only used mathematically to prevent confusion.

Radiation laws

Applying electromagnetism, thermodynamics, and quantum mechanics, to atoms and molecules, some laws of electromagnetic radiation and light are as follows.

• Stefan–Boltzmann law
• Planck's law of black-body radiation
• Wien's displacement law
• Radioactive decay law

Laws of chemistry

Chemical laws are those laws of nature relevant to chemistry. Historically, observations led to many empirical laws, though now it is known that chemistry has its foundations in quantum mechanics.

Quantitative analysis

The most fundamental concept in chemistry is the

The law of definite composition and the law of multiple proportions are the first two of the three laws of stoichiometry, the proportions by which the chemical elements combine to form chemical compounds. The third law of stoichiometry is the law of reciprocal proportions, which provides the basis for establishing equivalent weights for each chemical element. Elemental equivalent weights can then be used to derive atomic weights for each element.

More modern laws of chemistry define the relationship between energy and its transformations.

Reaction kinetics and equilibria

• In equilibrium, molecules exist in mixture defined by the transformations possible on the timescale of the equilibrium, and are in a ratio defined by the intrinsic energy of the molecules—the lower the intrinsic energy, the more abundant the molecule. Le Chatelier's principle states that the system opposes changes in conditions from equilibrium states, i.e. there is an opposition to change the state of an equilibrium reaction.
• Transforming one structure to another requires the input of energy to cross an energy barrier; this can come from the intrinsic energy of the molecules themselves, or from an external source which will generally accelerate transformations. The higher the energy barrier, the slower the transformation occurs.
• There is a hypothetical intermediate, or transition structure, that corresponds to the structure at the top of the energy barrier. The Hammond–Leffler postulate states that this structure looks most similar to the product or starting material which has intrinsic energy closest to that of the energy barrier. Stabilizing this hypothetical intermediate through chemical interaction is one way to achieve catalysis.
• All chemical processes are reversible (law of microscopic reversibility) although some processes have such an energy bias, they are essentially irreversible.
• The reaction rate has the mathematical parameter known as the
  rate constant. The Arrhenius equation gives the temperature and activation energy
  dependence of the rate constant, an empirical law.

Thermochemistry

• Dulong–Petit law
• Gibbs–Helmholtz equation
• Hess's law

Gas laws

• Raoult's law
• Henry's law

Chemical transport

• Fick's laws of diffusion
• Graham's law
• Lamm equation

Laws of biology

Ecology

• Competitive exclusion principle or Gause's law

Genetics

• Mendelian laws
  (Dominance and Uniformity, segregation of genes, and Independent Assortment)
• Hardy–Weinberg principle

Natural selection

Whether or not

• Arbia's law of geography
• Tobler's first law of geography
• Tobler's second law of geography

Geology

• Archie's law
• Buys-Ballot's law
• Birch's law
• Byerlee's law
• Principle of original horizontality
• Law of superposition
• Principle of lateral continuity
• Principle of cross-cutting relationships
• Principle of faunal succession
• Principle of inclusions and components
• Walther's law

Other fields

Some mathematical theorems and axioms are referred to as laws because they provide logical foundation to empirical laws.

Examples of other observed phenomena sometimes described as laws include the

law court. When we read Seneca's Natural Questions, and watch again and again just how he applies standards of evidence, witness evaluation, argument and proof, we can recognize that we are reading one of the great Roman rhetoricians of the age, thoroughly immersed in forensic method. And not Seneca alone. Legal models of scientific judgment turn up all over the place, and for example prove equally integral to Ptolemy's approach to verification, where the mind is assigned the role of magistrate, the senses that of disclosure of evidence, and dialectical reason that of the law itself.^[20]

The precise formulation of what are now recognized as modern and valid statements of the laws of nature dates from the 17th century in Europe, with the beginning of accurate experimentation and the development of advanced forms of mathematics. During this period,

Hobbes

(1588-1679).)

The distinction between natural law in the political-legal sense and law of nature or physical law in the scientific sense is a modern one, both concepts being equally derived from physis, the Greek word (translated into Latin as natura) for nature.^[24]

See also

• Empirical research
• Empirical statistical laws
• Formula
• List of laws
• Law (principle)
• Nomology
• Philosophy of science
• Physical constant
• Scientific laws named after people
• Theory

References

1. ^ "law of nature". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
2. ISBN 978-94-6209-497-0
   .
3. ^ "Definitions from". the NCSE. Retrieved 2019-03-18.
4. ISBN 978-0-309-11249-9
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5. ^ Gould, Stephen Jay (1981-05-01). "Evolution as Fact and Theory" (PDF). Discover. 2 (5): 34–37.
6. ISBN 0-19-866132-0
7. ^ "Law of nature". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
8. ^
   ISBN 978-0-671-79718-8
   .
9. ^
   ISBN 978-0-679-60127-2
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10. S2CID 122262641
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11. ^ Andrew S. C. Ehrenberg (1993), "Even the Social Sciences Have Laws", Nature, 365:6445 (30), page 385.(subscription required)
12. ISBN 0-201-02117-X
13. ^ Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1 (VHC Inc.) 0-89573-752-3
14. ISBN 0-07-084018-0
15. ISBN 0-691-03323-4
16. ISBN 0-07-084018-0
17. ^ Reed ES: The lawfulness of natural selection. Am Nat. 1981; 118(1): 61–71.
18. ^ ^{a b} Byerly HC: Natural selection as a law: Principles and processes. Am Nat. 1983; 121(5): 739–745.
19. ^ in Daryn Lehoux, What Did the Romans Know? An Inquiry into Science and Worldmaking (Chicago: University of Chicago Press, 2012), reviewed by David Sedley, "When Nature Got its Laws", Times Literary Supplement (12 October 2012).
20. ^ Sedley, "When Nature Got Its Laws", Times Literary Supplement (12 October 2012).
21. ISSN 0362-4331
    . Retrieved 2016-10-07. Isaac Newton first got the idea of absolute, universal, perfect, immutable laws from the Christian doctrine that God created the world and ordered it in a rational way.
22. ^ Harrison, Peter (8 May 2012). "Christianity and the rise of western science". ABC. Individuals such as Galileo, Johannes Kepler, Rene Descartes and Isaac Newton were convinced that mathematical truths were not the products of human minds, but of the divine mind. God was the source of mathematical relations that were evident in the new laws of the universe.
23. ^ "Cosmological Revolution V: Descartes and Newton". bertie.ccsu.edu. Retrieved 2016-11-17.
24. ^ Some modern philosophers, e.g. Norman Swartz, use "physical law" to mean the laws of nature as they truly are and not as they are inferred by scientists. See Norman Swartz, The Concept of Physical Law (New York: Cambridge University Press), 1985. Second edition available online [1].

Further reading

• ISBN 0-449-90738-4
  )
• Dilworth, Craig (2007). "Appendix IV. On the nature of scientific laws and theories". Scientific progress : a study concerning the nature of the relation between successive scientific theories (4th ed.). Dordrecht: Springer Verlag.
  ISBN 978-1-4020-6353-4
  .
• Francis Bacon (1620). Novum Organum.
• Hanzel, Igor (1999). The concept of scientific law in the philosophy of science and epistemology : a study of theoretical reason. Dordrecht [u.a.]: Kluwer.
  ISBN 978-0-7923-5852-7
  .
• Daryn Lehoux (2012). What Did the Romans Know? An Inquiry into Science and Worldmaking. University of Chicago Press. (
  ISBN 9780226471143
  )
• Nagel, Ernest (1984). "5. Experimental laws and theories". The structure of science problems in the logic of scientific explanation (2nd ed.). Indianapolis: Hackett.
  ISBN 978-0-915144-71-6
  .
• R. Penrose (2007).
  ISBN 978-0-679-77631-4
  .
• Swartz, Norman (20 February 2009). "Laws of Nature". Internet encyclopedia of philosophy. Retrieved 7 May 2012.

External links

Wikimedia Commons has media related to Scientific laws.

• Physics Formulary, a useful book in different formats containing many or the physical laws and formulae.
• Eformulae.com, website containing most of the formulae in different disciplines.
• Stanford Encyclopedia of Philosophy: "Laws of Nature" by John W. Carroll.
• Baaquie, Belal E. "Laws of Physics : A Primer". Core Curriculum, National University of Singapore.
• Francis, Erik Max. "The laws list".. Physics. Alcyone Systems
• Pazameta, Zoran. "The laws of nature".
  Committee for the scientific investigation of Claims of the Paranormal
  .
• The Internet Encyclopedia of Philosophy. "Laws of Nature" – By Norman Swartz
• "Laws of Nature", In Our Time, BBC Radio 4 discussion with Mark Buchanan, Frank Close and Nancy Cartwright (Oct. 19, 2000)